Vamshi Jandhyala
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Mathematical writing · Geometric Probability

Contents

63 chapters. Classical gems of geometric probability, with pictures and worked solutions.

  1. 1Three sticks forming a triangle
  2. 2The longest of three pieces
  3. 3Closer to centre than to any side
  4. 4A random quadratic with real roots
  5. 5The meeting problem
  6. 6Buffon’s coin (franc-carreau)
  7. 7Broken stick, second break on the longer piece
  8. 8A random quadratic with three uniform coefficients
  9. 9An n-gon from a broken stick
  10. 10Expected minimum distance to the boundary of a square
  11. 11Three friends at the café
  12. 12Two random chords crossing
  13. 13Expected distance between two points on a circle
  14. 14n random points on a circle, all in a semicircle
  15. 15Random chord longer than the radius
  16. 16Two random arcs on a circle
  17. 17Smallest gap among random points on a circle
  18. 18Intersections among n random chords
  19. 19Stevens’s covering theorem
  20. 20A random chord via a uniform midpoint
  21. 21Crofton’s expected chord of a disc
  22. 22Expected number of pieces cut by random chords
  23. 23Random chord through an inner concentric disc
  24. 24Random chord longer than the side of a regular n-gon
  25. 25Wendel’s theorem in the plane
  26. 26The acute-triangle probability
  27. 27Wendel’s theorem on the sphere
  28. 28Two random unit vectors on a sphere
  29. 29Five random points on a sphere
  30. 30A triangle in the disc that contains the centre
  31. 31Expected squared distance in a disc
  32. 32Expected area of a random rectangle
  33. 33Expected distance on a segment
  34. 34Expected distance from a random point in a square to its centre
  35. 35Disc line picking
  36. 36Distance between two points on the boundary of a square
  37. 37Expected maximum radius among n random points in a disc
  38. 38Expected range of n uniform random points on a segment
  39. 39Expected absolute y-coordinate in a disc
  40. 40Expected squared distance in a unit square
  41. 41Expected reciprocal distance from the centre of a disc
  42. 42Random walk from the centre of a unit square and cube
  43. 43Sylvester’s four-point problem in a triangle
  44. 44Expected area of a triangle in the unit square
  45. 45Sylvester’s four-point problem in a square
  46. 46Sylvester’s four-point problem in a disc
  47. 47Expected perimeter of a random triangle in a disc
  48. 48Expected lengths of the three broken-stick pieces
  49. 49Expected area of a random triangle in a disc
  50. 50Expected perimeter of a triangle inscribed in a circle
  51. 51Expected area of a triangle inscribed in a circle
  52. 52Visible lattice points
  53. 53Buffon’s needle
  54. 54Laplace’s extension of Buffon
  55. 55The lighthouse and the Cauchy distribution
  56. 56Three coprime integers
  57. 57Buffon’s noodle
  58. 58Cauchy’s shadow formula
  59. 59Expected distance in a ball
  60. 60Robbins’s constant
  61. 61Expected chord length on a sphere
  62. 62Topical Guide
  63. 63Bibliography